Partial Oxidation Synthesis of Prussian Blue Analogues for Thermo-Rechargeable Battery

Abstract

A thermo-rechargeable battery or tertiary battery converts thermal energy into electric energy via an electrochemical Seebeck coefficient. The manufacturing of the tertiary batteries requires a pre-oxidation step to align and optimize the cathode and anode potentials. The pre-oxidation step, which is not part of the secondary battery manufacturing process, makes the manufacturing of tertiary batteries complex and costly. To omit the pre-oxidation step, we used partially oxidized Prussian blue analogs, i.e., Na${}_x$Co[Fe(CN)${}_6$]${}_y$zH${}_2$O (Co-PBA) and Na${}_x$Ni[Fe(CN)${}_6$]${}_y$zH${}_2$O (Ni-PBA), as cathode and anode materials. The modified tertiary battery without the pre-oxidation step shows good thermal cyclability between 10 ${}^{\circ }$C and 50 ${}^{\circ }$C without detectable deterioration of the thermal voltage ($V_{\mathrm{cell}}$) and discharge capacity ($Q_{\mathrm{cell}}$).

1. Introduction

Energy harvesting from environmental heat is an important technology for our future society. Among the energy harvesting devices, thermo-rechargeable batteries [1,2,3,4,5,6,7,8] that can be charged by the warming/cooling of the device are promising. In thermo-rechargeable batteries, the temperature coefficient ($\alpha$≡ dV/dT) of redox potential (V) is different between the cathode and anode. The thermo-rechargeable batteries convert thermal energy into electric energy between the low ($T_{\mathrm{L}}$) and high ($T_{\mathrm{H}}$) heat sources, analogously to a heat engine. Hereafter, we call the thermo-rechargeable battery composed of solid active materials a tertiary battery, because it has the same device configuration as an ion secondary battery. With an increase in the device temperature from $T_{\mathrm{L}}$ to $T_{\mathrm{H}}$, cell voltage ($v_{\mathrm{cell}}$) increases from 0 V to $V_{\mathrm{cell}}$ (≥0). The battery produces electric energy through the discharge processes at $T_{\mathrm{H}}$. With a decrease in the device temperature to $T_{\mathrm{L}}$, $v_{\mathrm{cell}}$ decreases from 0 V to $-V_{\mathrm{cell}}$. The battery produces electric energy through the discharge processes at $T_{\mathrm{L}}$. A tertiary battery can generate electrical energy from daily temperature cycles such as day and night or sunlight and shade. Unlike a thermo-electric conversion device that converts the temperature difference within the device into electric energy, the tertiary battery converts the temperature change of the device into electric energy. Like batteries, tertiary batteries are mobile because the device does not need to be in contact with a heat source. Thus, a tertiary battery is a device that has both mobility like a dry battery and independence from a power grid like a thermoelectric conversion device.

The thermal voltage ($V_{\mathrm{cell}}$) and discharge capacity ($Q_{\mathrm{cell}}$) are important performance parameters of a battery. The thermal voltage is expressed by electrode parameters as $V_{\mathrm{cell}}^{\mathrm{cal}}$ = (${\alpha }^{+}$−${\alpha }^{-}$)($T_{\mathrm{H}}$−$T_{\mathrm{L}}$), where ${\alpha }^{+}$ (${\alpha }^{-}$) is $\alpha$ of cathode (anode) material. We emphasize that cathode and anode potentials should be the same when they are assembled into a tertiary battery. We called this potential adjustment step the pre-oxidization step. Generally, the voltage of a battery decreases as the extracted charge increases. The discharge capacity of a tertiary battery is defined as the capacity at which the voltage drop is equal to $V_{\mathrm{cell}}$. Since $V_{\mathrm{cell}}$ of the tertiary battery is relatively small (several tens mV), the charge coefficient ($\beta$ ≡ −dV/dq, where q is capacity per unit weight of active material) of V can be regarded as constant. Considering the weight ratio (r = $\frac{m^{+}}{m^{+}+m^{-}}$) of the cathode ($m^{+}$) and anode ($m^{-}$) active materials, the discharge capacity per unit weight of total active material is expressed by electrode parameters as $Q_{\mathrm{cell}}^{\mathrm{cal}}$ = $\frac{V_{\mathrm{cell}}^{\mathrm{cal}}}{\frac{{\beta }_{+}}{r}+\frac{{\beta }_{-}}{1-r}}$, where ${\beta }^{+}$ (${\beta }^{-}$) and is $\beta$ of the cathode (anode) material [7,8]. The formula tells us that $Q_{\mathrm{cell}}^{\mathrm{cal}}$ is maximized when the cathode and anode potentials are optimized so that $\frac{{\beta }_{+}}{r}+\frac{{\beta }_{-}}{1-r}$ is minimized. The adjustment of the cathode and anode potential is also performed at the pre-oxidization step. Thus, the pre-oxidization step is indispensable for the fabrication of a tertiary battery.

Prussian blue analogs, Na${}_x$M[Fe(CN)${}_6$]${}_y$zH${}_2$O (M-PBAs; M is divalent metal element), are promising electrode materials for lithium-ion secondary battery (LIBs) [9,10,11,12,13,14], sodium-ion secondary batteries (SIBs) [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34], and tertiary batteries [4,5,6]. M-PBAs show a jungle-gym-type framework structure; the intersection points and edges are metal elements and cyano groups, respectively. The framework can accommodate Na${}^{+}$ and H${}_2$O molecules. Importantly, the accommodated Na${}^{+}$ can be electrochemically moved in and out accompanying reduction and oxidation of the framework. In the reduced state of M-PBA, Fe and M takes a divalent state and each nanopore accommodates one Na${}^{+}$. In the oxidization process, the valence of Fe and/or M increases to trivalent with removing Na${}^{+}$ from the nanopore. Most of M-PBA in reduced state show face-centered cubic (fcc) ($Fm\overline{3}m$; Z = 4) or trigonal ($R\overline{3}m$; Z = 3) structures [35,36,37]. As active materials for tertiary batteries, M-PBAs have three advantages. First, they can be synthesized as precipitates from aqueous solutions. In addition, they can be synthesized at a low cost since they do not contain rare elements. Second, M-PBA and its oxidants are stable in air and in aqueous solutions. In particular, a reversible redox reaction is possible in aqueous solution. Third, M-PBA shows positive and negative $\alpha$. Therefore, a tertiary battery can be composed of M-PBA with positive $\alpha$ and M-PBA with negative $\alpha$. The $\alpha$ values are negative in Mn-, Ni-, Cu-, and Zn-PBAs and are positive in Fe- and Co-PBAs [38]. Among M-PBAs, Co-PBA ($\alpha$ = +0.57 mV/K [38]) and Ni-PBA (−0.42 mV/K [38]) are prototypical combinations as cathode and anode materials for tertiary batteries [5], because their cycle stability is good as compared with the other combination.

The conventional tertiary battery is fabricated by four steps, i.e., (i) material synthesis, (ii) electrode fabrication, (iii) pre-oxidization of electrodes, and (iv) battery assembly (Figure 1a). Unlike LIBs/SIBs (Figure 1c), the manufacturing of tertiary batteries requires the (iii) pre-oxidation step for the following two reasons. First, the cathode and anode potentials should be the same before they are assembled into a tertiary battery, Second, the potentials should be optimized so that $\frac{{\beta }_{+}}{r}+\frac{{\beta }_{-}}{1-r}$ is minimized and $Q_{\mathrm{cell}}$ is maximized. In the as-grown reduced state, the potential gradient ($\beta$) aganist charge is significantly large because the charge curve steeply increases with q. The pre-oxidation step flattens the potential gradient. The pre-oxidization step, which is performed by electrochemical oxidization of the electrodes, makes the manufacture of tertiary batteries complex and costly. If we directly synthesize partially oxidized M-PBA, we can omit the pre-oxidation step from the manufacturing process (Figure 1b). We will call such a battery a modified tertiary battery.

In this study, we performed partial oxidization synthesis of Co-PBA and Ni-PBA to omit the pre-oxidization step in the manufacturing of the tertiary batteries. The modified tertiary battery without the pre-oxidation step shows good thermal cyclability between $T_{\mathrm{L}}$ (=10 ${}^{\circ }$C) and $T_{\mathrm{H}}$ (=50 ${}^{\circ }$C) without detectable deterioration of $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$. The $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ values are close to the ideal $V_{\mathrm{cell}}^{\mathrm{cal}}$ and $Q_{\mathrm{cell}}^{\mathrm{cal}}$ values. By incorporating the partial oxidation synthesis, the tertiary battery can be prepared by the same processes as LIBs/SIBs and will possess an industrial advantage.

2. Materials and Methods

2.1. Partial Oxidation Synthesis

Partial oxidization synthesis of Co-PBAs and Ni-PBAs, i.e., Na${}_x$Co[Fe(CN)${}_6$]${}_y$zH${}_2$O and Na${}_x$Ni[Fe(CN)${}_6$]${}_y$zH${}_2$O, were performed by dipping solution A into solution B at a speed of 100 mL/h at 35 °C Solution A was an aqueous solution containing 10 mM MCl${}_2$ and 4 M NaCl. Solution B was an aqueous solution containing 10(1 −$n_{\mathrm{Fe}3+}^0$) mM Na${}_4$[Fe(CN)${}_6$], 10$n_{\mathrm{Fe}3+}^0$ mM K${}_3$[Fe(CN)${}_6$], and 4 M NaCl. The degree of the oxidization was controlled by the initial Fe${}^{3+}$ ratio ($n_{\mathrm{Fe}3+}^0$). In preparation of solution B ($n_{\mathrm{Fe}3+}^0$> 0.0), K${}^{+}$ in the solution was removed by an ion-exchange method. The aqueous solution containing 10(1 −$n_{\mathrm{Fe}3+}^0$) mM Na${}_4$[Fe(CN)${}_6$] and 10$n_{\mathrm{Fe}3+}^0$ mM K${}_3$[Fe(CN)${}_6$] was stirred for 3 h at 50 ${}^{\circ }$C with 50$n_{\mathrm{Fe}3+}^0$ mM Ni-PBA ($n_{\mathrm{Fe}3+}^0$ = 0.0) powder. In this process, K${}^{+}$ in solution exchanges with Na${}^{+}$ in Ni-PBA. The ion-exchanged solutions were filtered and added 4 M NaCl.

The molar ratios of Fe and K to Co/Ni (

Table 1. Molar ratios of Fe and K to Co/Ni of partially-oxidized Co-PBAs and Ni-PBAs. $n_{\mathrm{Fe}3+}^0$ are the initial Fe${}^{3+}$ ratio. Quantification of Na moles was difficult due to the weak intensity of the EDX signal.

	${\mathit{n}}_{\mathbf{Fe}3+}^0$	Co/Ni	Fe	K
Co-PBA	0.0	1	0.88	below the limits of detection
Co-PBA	0.6	1	0.84	0.01
Co-PBA	0.8	1	0.85	0.01
Co-PBA	1.0	1	0.88	0.02
Ni-PBA	0.0	1	0.83	below the limits of detection
Ni-PBA	0.5	1	0.86	below the limits of detection
Ni-PBA	0.9	1	0.91	below the limits of detection
Ni-PBA	1.0	1	1.00	0.05

) were determined using an SEM-EXS machine (JST-IT2000; JEOL, Ltd., Akishima, Japan). The molar ratios of C, N, and H (

Table 2. Weight percent of C, N, and H of partially-oxidized Co-PBAs and Ni-PBAs. The experimental error was less than 0.1 wt%. $n_{\mathrm{Fe}3+}^0$ are the initial Fe${}^{3+}$ ratio.

	${\mathit{n}}_{\mathbf{Fe}3+}^0$	C (wt%)	N (wt%)	H (wt%)
Co-PBA	0.0	19.07	22.07	1.78
Co-PBA	0.6	19.08	21.75	1.96
Co-PBA	0.8	19.07	21.85	1.90
Co-PBA	1.0	19.43	22.17	1.77
Ni-PBA	0.0	17.88	20.65	2.25
Ni-PBA	0.5	18.23	20.57	2.11
Ni-PBA	0.9	18.42	20.74	2.26
Ni-PBA	1.0	18.49	20.86	2.23

) were evaluated by a CHN analyzer (elementar; UNICUBE). The water content (z) was evaluated by z = 3y$\frac{m_{\mathrm{H}}}{m_{\mathrm{CN}}}$, where $m_{\mathrm{CN}}$ and $m_{\mathrm{H}}$ are the molar ratio of CN and H. Thus, the obtained chemical compositions of Co-PBAs and Ni-PBAs are listed in

Table 3. Chemical composition and lattice constant of partially-oxidized Co-PBAs and Ni-PBAs. $n_{\mathrm{Fe}3+}^0$ and p are the initial Fe${}^{3+}$ ratio and degree of the oxidization, respectively. Fcc means face-centered-cubic cell. Subscript H means hexagonal setting.

	${\mathit{n}}_{\mathbf{Fe}3+}^0$	p	Composition	Structure	Lattice Constant (Å)
Co-PBA	0.0	0.00	Na${}_{1.52}$Co[Fe(CN)${}_6$]${}_{0.88}$2.96H${}_2$O	trigonal	$a_{\mathrm{H}}$ = 7.428(3), $c_{\mathrm{H}}$ = 17.57(1)
Co-PBA	0.6	0.06	Na${}_{1.28}$Co[Fe(CN)${}_6$]${}_{0.84}$3.14H${}_2$O	fcc	a = 10.247(4)
Co-PBA	0.8	0.07	Na${}_{1.31}$Co[Fe(CN)${}_6$]${}_{0.85}$3.07H${}_2$O	fcc	a = 10.206(3)
Co-PBA	1.0	0.10	Na${}_{1.37}$Co[Fe(CN)${}_6$]${}_{0.88}$2.91H${}_2$O	fcc	a = 10.181(3)
Ni-PBA	0.0	0.00	Na${}_{1.32}$Ni[Fe(CN)${}_6$]${}_{0.83}$3.77H${}_2$O	fcc	a = 10.290(6)
Ni-PBA	0.5	0.10	Na${}_{1.35}$Ni[Fe(CN)${}_6$]${}_{0.86}$3.65H${}_2$O	fcc	a = 10.297(4)
Ni-PBA	0.9	0.16	Na${}_{1.57}$Ni[Fe(CN)${}_6$]${}_{0.93}$4.17H${}_2$O	fcc	a = 10.291(3)
Ni-PBA	1.0	0.15	Na${}_{1.85}$Ni[Fe(CN)${}_6$]${}_{1.00}$4.41H${}_2$O	fcc	a = 10.272(4)

. The Fe concentration (y) ranged from 0.83 to 1.00. The molar ratio of K is less than 0.05. The Na concentration (x) was calculated from y and the degree of oxidization (p; vide infra) assuming charge neutrality. In the charge process of Co-PBA, Co${}^{2+}$ is oxidized first and then Fe${}^{3+}$ is oxidized [18]. Then, we formally obtained the relation x = (1−p)(4y−2). In the charging process of Ni-PBA, only Fe${}^{2+}$ is oxidized [14]. Then, we obtained the relation x = (1−p)y + (3y−2).

Figure 2 shows the X-ray diffraction (XRD) patterns of (a) Co-PBA and (b) Ni-PBA against $n_{\mathrm{Fe}3+}^0$. The XRD patterns were obtained using an XRD machine (MultiFlex; Rigaku) at room temperature. The X-ray source was the CuK$\alpha$ line. Except for Co-PBA ($n_{\mathrm{Fe}3+}^0$ = 0.0) (Figure 2a), all the diffraction peaks can be indexed in the fcc cell. For Co-PBA ($n_{\mathrm{Fe}3+}^0$ = 0.0), all the diffraction peaks can be indexed in the trigonal cell (hexagonal setting). With the use of the Rietan-FP program [39], we refined the lattice constants. The obtained lattice constants are listed in Table 3.

2.2. Electrode Fabrication

Paste-type electrodes of Co-PBAs and Ni-PBAs were prepared as follows. First, M-PBA, acetylene black, and PVDF were well mixed with DMF at a ratio of 7:2:1. Then, the slurry was applied on an indium tin oxide (ITO) electrode and dried in a vacuum at 60 °C. The slurry was carefully applied by hand so that the thickness was uniform. The area of the electrode was 1.0 cm${}^2$. To reduce the weight error, the weights of the electrodes with and without slurry were measured for each electrode. The weight of the active material, which is 70% of the slurry weight, was evaluated from the difference in the electrode weights.

We fabricated a three-pole beaker-type cell and measured charge and discharge curves with a potentiostat (HJ1001SD8; HokutoDENKO) at a rate of 0.3 C. The working, counter, and referential electrodes of the cell were the sample electrode, Pt, and Ag/AgCl standard electrodes, respectively. The electrolyte was an aqueous solution containing 17 mol/kg NaClO${}_4$. We used theoretical capacity in the fully discharged state to define the C rate. The ideal capacities of Co-PBA in the fully discharged state are 122, 113, 116, and 123 mAh/g at $n_{\mathrm{Fe}3+}^0$ = 0.0, 0.6, 0.8, and 1.0, respectively. The currents at 1 C for a 1 mg electrode are 1.32, 1.26, 1.28, and 1.35 mA at $n_{\mathrm{Fe}3+}^0$ = 0.0, 0.6, 0.8, and 1.0, respectively. The ideal capacities of Ni-PBA in the fully discharged state are 67, 68, 68, and 68 mAh/g at $n_{\mathrm{Fe}3+}^0$ = 0.0, 0.5, 0.9, and 1.0, respectively. The currents at 1 C for a 1 mg electrode are 0.72, 0.73, 0.67, and 0.63 mA at $n_{\mathrm{Fe}3+}^0$ = 0.0, 0.5, 0.9, and 1.0, respectively. The potential range was from 1.1 and 0.3 V (vs. Ag/AgCl).

2.3. Battery Assembly

The Co-PBA/Ni-PBA tertiary batteries were two-pole beaker-type cell. The cathode, anode, and electrolyte were the Co-PBA electrode, Ni-PBA electrode, and aqueous solution containing 17 mol/kg NaClO${}_4$, respectively. We note that $Q_{\mathrm{cell}}^{\mathrm{cal}}$ (= $\frac{V_{\mathrm{cell}}^{\mathrm{cal}}}{\frac{{\beta }_{+}}{r}+\frac{{\beta }_{-}}{1-r}}$) is maximized when the $\beta$ values of the cathode and anode are minimized. Figure 3 shows $\beta$ of (a) Ni-PBA and (b) Co-PBA electrodes against V. The $\beta$ values of Co-PBA and Ni-PBA are minimized at V≈ 510 mV (Figure 3). As will be described in the next section, Co-PBA at $n_{\mathrm{Fe}3+}^0$ = 1.0 and Ni-PBA at 1.0 satisfy this condition.

We assembled a Co-PBA($n_{\mathrm{Fe}3+}^0$ = 1.0)/Ni-PBA(1.0) tertiary battery (modified cell) without the pre-oxidation step (Figure 1b). For comparison, we assembled a conventional Co-PBA($n_{\mathrm{Fe}3+}^0$ = 0.0)/Ni-PBA(0.0) tertiary battery with the pre-oxidation step (Figure 1a). The pre-oxidization was performed with the use of the three-pole beaker-type cell. In

Table 4. Parameters of cathodes and anodes for the modified and conventional cells. ${\beta }^{+}$ (${\beta }^{-}$), $m^{+}$ ($m^{-}$), $R^{+}$ ($R^{-}$), and $V^{+}$ ($V^{-}$) are the charge coefficient ($\beta$≡−dV/dq) of V, the weight of active material, internal resistivity, and initial potential (vs. Ag/AgCl) of cathode (anode), respectively. The electrolyte was 17 kg/L NaClO${}_4$.

Cathode	Material	${\beta }^{+}$ ($\Omega$g/h)	$m^{+}$ (mg)	$V^{+}$ (mV)	${\alpha }^{+}$ (mV/K)	$R^{+}$ (k$\Omega$)
modified cell	Co-PBA	4.1	0.70	510	0.57 [38]	0.40
conventional cell	Co-PBA	5.0	0.26	520	0.57 [38]	0.71
Anode	Material	${\beta }^{-}$ ($\Omega$g/h)	$m^{-}$ (mg)	$V^{-}$ (mV)	${\alpha }^{+}$ (mV/K)	$R^{-}$ (k$\Omega$)
modified cell	Ni-PBA	7.6	0.27	510	−0.42 [38]	0.74
conventional cell	Ni-PBA	5.3	0.13	520	−0.42 [38]	0.80

, we summarized the electrode parameters for the modified and conventional cells.

2.4. Thermal Cycle Measurement

The thermal cycle properties of the tertiary batteries were investigated between $T_{\mathrm{L}}$ (=10 °C) and $T_{\mathrm{H}}$ (=50 °C). The temperature (T) of the battery was controlled in a thermobath. To stabilize the potential of the electrodes, the cell was connected to an external resistor and held for several hours at $T_{\mathrm{L}}$ (=10 °C) before the measurement. The thermal cycle consists of four processes, (i) heating, (ii) discharge at $T_{\mathrm{H}}$, (iii) cooling, and (iv) discharge at $T_{\mathrm{L}}$. In the (i) heating process, T was slowly increased from $T_{\mathrm{L}}$ to $T_{\mathrm{H}}$ under open circuit conditions. Then, the cell was kept at $T_{\mathrm{H}}$ for 15 min to stabilize the potential. At (ii) $T_{\mathrm{H}}$, the discharge process was investigated at 0.03 C until the $v_{\mathrm{cell}}$ reached 0 mV. The C rate was defined by the theoretical capacity of the Co-PBA electrode in the fully discharged state. The currents at 1 C for a 1 mg Co-PBA electrode are 1.35 and 1.32 mA for the modified and conventional cells, respectively. In the (iii) cooling process, T was slowly decreased to $T_{\mathrm{L}}$. Then, the cell was kept at $T_{\mathrm{L}}$ for 15 min. At (iv) $T_{\mathrm{L}}$, the discharge process was investigated at 0.03 C until $v_{\mathrm{cell}}$ reaches 0 mV. The thermal cycle measurements were repeated five times.

3. Results and Discussion

3.1. Degree of Oxidization

Figure 4a shows first (red) and second (blue) charge curves of partially oxidized Co-PBA at 0.3 C and 25 °C against $n_{\mathrm{Fe}3+}^0$. At $n_{\mathrm{Fe}3+}^0$ = 0.0, the first and second charge curves coincide with each other. The charge curves show a two-plateau structure at ∼1.0 V and ∼0.6 V. The upper and lower plateaus are ascribed to the redox process of [Fe(CN)${}_6$]${}^{4-}$/[Fe(CN)${}_6$]${}^{3-}$ and Co${}^{3+}$/Co${}^{2+}$, respectively [18]. The observed charge capacity (=113 mAh/g) is close to the ideal value (=122 mAh/g). At $n_{\mathrm{Fe}3+}^0$ = 1.0, the first charge capacity (=98 mAh/g) is smaller than second charge capacity (=109 mAh/g), reflecting the the partial oxidization of Co-PBA. The first charge curve is overlapped with the second charge curve if the curve is shifted by 11 mAh/g. The partially oxidized Co-PBA ($n_{\mathrm{Fe}3+}^0$ = 1.0) shows initial potential of 0.50 V and is suitable for the cathode material for the Co-PBA/Ni-PBA tertiary battery. Similar behaviors are observed at $n_{\mathrm{Fe}3+}^0$ = 0.6 and 0.8. Figure 5a shows the first (red) and second (blue) charge curves of partially oxidized Co-PBA ($n_{\mathrm{Fe}3+}^0$ = 0.8) at 0.3 C and 25 ${}^{\circ }$C. The first and second discharge curves coincide with each other even though the first and second charge curves (Figure 4a) differ significantly. Thus, the partially oxidized Co-PBA shows stable charge properties.

Figure 4b shows the first and second charge curves of partially oxidized Ni-PBA at 0.3 C and 25 ${}^{\circ }$C against $n_{\mathrm{Fe}3+}^0$. At $n_{\mathrm{Fe}3+}^0$ = 0.0, the first and second charge curves coincide with each other. The charge curves show a single plateau structure. The plateau is ascribed to the redox process of [Fe(CN)${}_6$]${}^{4-}$/[Fe(CN)${}_6$]${}^{3-}$ [14]. The observed charge capacity (=54 mAh/g) is close to the ideal value (=67 mAh/g). At $n_{\mathrm{Fe}3+}^0$ = 1.0, the first charge capacity (=62 mAh/g) is smaller than the second charge capacity (=73 mAh/g), reflecting the partial oxidization of Ni-PBA. The first charge curve is overlapped with the second charge curve if the curve is shifted by 11 mAh/g. The partially oxidized Ni-PBA ($n_{\mathrm{Fe}3+}^0$ = 1.0) shows an initial potential of 0.53 V and is suitable for the anode material for the Co-PBA/Ni-PBA tertiary battery. Similar behaviors are observed at $n_{\mathrm{Fe}3+}^0$ = 0.5 and 0.9. Figure 5b shows the first (red) and second (blue) charge curves of partially oxidized Ni-PBA ($n_{\mathrm{Fe}3+}^0$ = 0.9) at 0.3 C and 25 °C. The first and second discharge curves coincide with each other even though the first and second charge curves (Figure 4b) differ significantly. Thus, the partially oxidized Co-PBA shows stable charge properties.

Here, we define the degree of oxidization (p) as the ratio between the oxidization charge and the observed capacity of the fully reduced compound. When the active material is partially oxidized, the initial capacity is the difference between the observed capacity in the fully reduced compound and the oxidation charge. The second cycle capacity corresponds to the observed capacity in the fully reduced compound. Then, p is expressed as 1- (initial capacity)/(second cycle capacity). Figure 6 shows the p values of (a) Co-PBA and (b) Ni-PBA against $n_{\mathrm{Fe}3+}^0$. In both the compounds, the p value linearly increases with an increase in $n_{\mathrm{Fe}3+}^0$, as indicated by straight lines. In Ni-PBA, p is proportional to the concentration of Fe${}^{3+}$ in the as-grown sample. Then, the observed proportional relationship indicates that the as-grown Ni-PBA contains Fe${}^{3+}$ in proportion to $n_{\mathrm{Fe}3+}^0$. In Co-PBA, p is not directly proportional to the concentration of Fe${}^{3+}$ in the as-grown sample, since charge transfer from Co${}^{2+}$ to Fe${}^{3+}$ occurs within the sample. The observed proportional relationship is probably because the amount of charge transfer is proportional to the concentration of Fe${}^{3+}$ incorporated during crystal growth.

3.2. Thermal Cycle Properties

Figure 7 shows the first thermal cycle of the modified Co-PBA/Ni-PBA tertiary battery between $T_{\mathrm{L}}$ (=10 °C) and $T_{\mathrm{H}}$ (=50 °C). The modified cell was fabricated without the pre-oxidization step (Figure 1b). In the (i) heating process, $v_{\mathrm{cell}}$ increases from 0 mV to 40 mV (=$V_{\mathrm{cell}}$). In the (ii) discharge at $T_{\mathrm{H}}$, $v_{\mathrm{cell}}$ linearly decreases from $V_{\mathrm{cell}}$ to 0 mV. In the (iii) cooling process, $v_{\mathrm{cell}}$ decreases from 0 mV to −40 mV (=−$V_{\mathrm{cell}}$). In the (iv) discharge at $T_{\mathrm{L}}$, $v_{\mathrm{cell}}$ linearly increases from −$V_{\mathrm{cell}}$ to 0 mV. Figure 7b shows the thermal cycle of the modified cell between $T_{\mathrm{L}}$ (=10 °C) and $T_{\mathrm{H}}$ (=50 °C). We observed no serious deterioration of $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ up to five cycles. For comparison, we show in Figure 7c the thermal cyclability of the conventional cell.

Figure 8a shows $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ of the modified cell against cycle number. The $Q_{\mathrm{cell}}$ value was evaluated from the discharge time. $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ are nearly constant against cycle number. We show in Figure 8b $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ of the conventional cell against cycle number. $V_{\mathrm{cell}}$ are nearly constant against cycle number. $Q_{\mathrm{cell}}$, however, gradually decreases with an increase in cycle number. Thus, the capacity cyclability of the modified cell (Figure 8a) is improved as compared with that of the conventional cell (Figure 8b).

Takahara et al. [5] fabricated two tertiary batteries made of thin films of M-PBA, i.e., Na${}_x$Co[Fe(CN)${}_6$]${}_{0.90}$/Na${}_x$Co[Fe(CN)${}_6$]${}_{0.71}$ and Na${}_x$Co[Fe(CN)${}_6$]${}_{0.90}$/Na${}_x$Ni[Fe(CN)${}_6$]${}_{0.68}$ cells, and investigated their thermal cyclabilities between 10 °C and 50 °C. They found that the capacity cyclability is much improved in the latter cell and ascribed the improvement to better capacity cyclability of the Na${}_x$Ni[Fe(CN)${}_6$]${}_{0.68}$ anode at 50 °C. Then, the better capacity cyclability observed in the modified cell (Figure 8a) is considered to be ascribed to better capacity cyclability of the active materials used in the modified cell. With use of the Scherrer’s equation, we evaluated the grain sizes of Co-PBAs and Ni-PBAs used in the modified and conventional cells. The grain sizes of the active materials in the modified cell were 20 nm (Co-PBA, $n_{Fe3+}^0$ = 1.0) and 10 nm (Ni-PBA, $n_{Fe3+}^0$ = 1.0) whereas those in the conventional cell were 100 nm (Co-PBA, $n_{Fe3+}^0$ = 0.0) and 10 (Ni-PBA, $n_{Fe3+}^0$ = 0.0). The smaller grain size in the modified cell is advantageous for the stable electrochemical process, and hence, the capacity cyclability.

3.3. Comparison between Observed and Calculated $V_{\mathrm{cell}}$ Anmd $Q_{\mathrm{Cell}}$

Finally, let us compare the observed $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ values with the calculated values, i.e., $V_{\mathrm{cell}}^{\mathrm{cal}}$ [= (${\alpha }^{+}-{\alpha }^{-}$)($T_{\mathrm{H}}-T_{\mathrm{L}}$)] and $Q_{\mathrm{cell}}^{\mathrm{cal}}$ [=$\frac{{\alpha }^{+}-{\alpha }^{-}}{\frac{{\beta }^{+}}{r}+\frac{{\beta }^{-}}{1-r}}$($T_{\mathrm{H}}-T_{\mathrm{L}}$)] [7,8]. The electrode parameters, ${\alpha }^{+}$ (${\alpha }^{-}$), ${\beta }^{+}$ (${\beta }^{-}$), and r ($\frac{m^{+}}{m^{+}+m^{-}}$), are listed in Table 4. The temperature difference (=$T_{\mathrm{H}}-T_{\mathrm{L}}$) is 40 K. In the modified cell, $V_{\mathrm{cell}}^{\mathrm{cal}}$ and $Q_{\mathrm{cell}}^{\mathrm{cal}}$ are 39.6 mV and 1.2 mAh/g, respectively. The observed $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ (Figure 8a) are close to the calculated values. In the conventional cell, $V_{\mathrm{cell}}^{\mathrm{cal}}$ and $Q_{\mathrm{cell}}^{\mathrm{cal}}$ are 39.6 mV and 1.7 mAh/g, respectively. The observed $V_{\mathrm{cell}}$ values (Figure 8b) are close to the calculated value (=39.6 mV). In the conventional cell (Figure 8b), $Q_{\mathrm{cell}}$ at $T_{\mathrm{H}}$ is much smaller than $Q_{\mathrm{cell}}$ at $T_{\mathrm{L}}$ even though their average is around the calculated value (=1.7 mAh/g). We note that the difference in $Q_{\mathrm{cell}}$ between $T_{\mathrm{H}}$ and $T_{\mathrm{L}}$ has a serious effect on the long-term thermal stability. The cathode (anode) material in a tertiary battery is reduced (oxidized) at $T_{\mathrm{H}}$ and oxidized (reduced) at $T_{\mathrm{L}}$. Therefore, if $Q_{\mathrm{cell}}$ differs significantly, the Na${}^{+}$ concentration (x) of the electrode materials will change as the cycles number increases.

4. Conclusions

We proposed a modified manufacturing method for the Co-PBA/Ni-PBA tertiary battery free from the pre-oxidation step. Especially, the capacity cyclability of the modified cell between $T_{\mathrm{L}}$ (=10 °C) and $T_{\mathrm{H}}$ (=50 °C) is better than that of the conventional one. The observed $V_{\mathrm{cell}}$ and $Q_{\mathrm{cell}}$ are reproduced by the calculation based on the electrode parameters. Our method is applicable to other electrode materials and enables the manufacturing of tertiary batteries in the same steps as the commercialized LIBs. From the viewpoint of the manufacturing process and manufacturing costs, the tertiary battery is considered to be one of the promising energy harvesting devices.